changchang xi - universität bielefeld · changchang xi (¨ ¨ ¨ ~ ~ ~ ) ... let us look at the...

Main purpose ofthis talk

Happel-CPS-Rickard-KellerTheorem forclassical tiltingmodules

Modern tiltingtheory

Exact pairs

Homologicalsubcategories

Counterexamplesand openquestions

The 15. International Conference on Representations of Algebras, Bielefeld, August 13-17, 2012

Infinite dimensional tilting modules, homologicalsubcategories and recollements

Changchang Xi (¨̈̈��~~~)

Capital Normal UniversityBeijing, China

12:00-12:20, August 17, 2012




Exact pairs



Some groups in this direction

Finite dimensional tilting theory is well-known.

Let us look at the infinite dimensional titling theory.There are several groups in this area:

(1) Italy [Angeleri-Huegel, Bazzoni, Colpi, Mantese, Pavarin, Tonolo, ...]

(2) Spain [Herbera, Nicolas, Sanchez, Saorin, ...]

(3) Czech [Stovicek, Trifaj, ...]

(4) Germany [Koenig and his group, ...]

(5) · · ·




Exact pairs



Main aim

Some developments on infinitely generated tiltingmodules in terms of derived module categories.But restrict to [3] and [4]

[1 ] Proc. London Math. Soc. 104(2012) 959-996.

[2 ] arXiv:1107.0444 [Stratifications of derived categories from tilting

modules over tame hereditary algebras.]

[3 ] arXiv:1203.5168v2 [Homological ring epimorphisms and

recollements from exact pairs, I.][4 ] arXiv:1206.0522 [ Ringel modules and homological subcategories]

These are joint works with Hongxing Chen.




Exact pairs



Notations

A : ring with 1A-Mod: cat. of all left R-modulesadd(M): summands of f. dir. sums of MAdd(M): summands of dir. sums of MD(A): derived cat. of A (or A-Mod)




Exact pairs



Definition of tilting modules

Definitionn-tilting module AT :

(1) pdA(T)≤ n: P• −→ T → 0,

(2) ExtiA(T,T(I)) = 0 for all i > 0 and all set I,

(3) exact seq.: 0→ AA→ T0 → ··· → Tn → 0, Tj ∈ Add(T).

good if Ti ∈ add(T).classical if good and f.g.

Define B := EndA(T)




Exact pairs



Classical tilting and der. equivalences

Happel Theorem

Theorem

AT : classical n-tilting, =⇒ D(A)'D(B).

Note:

Derived invaraints

No new triangulated categories




Exact pairs



Classical tilting and der. equivalences

Natural question:

Is Happel Theorem still true?

AT : inf. g. tilting =⇒ D(A)'D(B)?




Exact pairs



Bazzoni’s Answer

Theorem (Bazzoni, Bazzoni-Mantese-Tonolo)

AT : n-tilting =⇒ D(A): subcategory or quotient ofD(B).

In fact: D(B)/Ker(T⊗LB−) ' D(A)

Note:

New triangulated categories

No derived invariants




Exact pairs



Happel Theorem for general tilting modules

What should bethe corresp. Happel Theoremfor inf. g. tilt. mod.s?




Exact pairs



Definition of homological ring epimorphisms

Definition

A ring epimorphism λ : R→ S is calledhomological if TorR

j (S,S) = 0 for j > 0.

Or equivalently, the restriction functorD(λ∗) : D(S)→D(R) is fully faithful.

Reference: Geigle-Lenzing: J. Algebra 144(1991)273-343.




Exact pairs



For n=1 case

Theorem

AT : good tilt., proj.dim ≤ 1, =⇒∃ homolog. ringepi. B→ C and recollement:

D(C) // D(B)j! //

ee

yy

D(A)ee

yy

T : classical, ⇒ C = 0, Happel Theorem.

j! := T⊗LB −,Ker(j!)'D(C).

C: universal localization of B.




Exact pairs



Recollements instead of derived equivalences

n = 1 indicates:

For inf. g. tilt. mod.s, Happel Theorem should be arecollement of der. module categories.

Immediate question:

Is the above theorem true for n-tilting mod.s ?




Exact pairs



General case

This is a difficult question!

It is related to questions:(1) When is a universal localization homological?(2) When is a full triangulated subcategory T ofD(B) realisable as a der. module cat.?

Question (1) is a very general, old question. Question (2) may be new,

but also very general. We shall consider a special case which is related

to inf. g. tilting modules.




Exact pairs



Definition of exact pairs of ring homomorphisms

λ : R−→ S, µ : R−→ T are ring hom.s.

The pair (λ,µ) is called exact if

0→ R→ S⊕T

(λ⊗1−1⊗µ

)−→ S⊗R T → 0

is exact as R-R-bimodules.




Exact pairs



• The coproduct StR T :

R

µ��

λ // S

ρ

��T

ϕ // StR T.

• Define a universal localization (due to Schofield):

θ : B :=(

S S⊗R T0 T

)−→

(StR T StR TStR T StR T

)=: C,

(s1 s2⊗ t20 t1

)7→

((s1)ρ (s2)ρ(t2)ϕ

0 (t1)ϕ

)




Exact pairs



This θ is related to inf. g. tilting modules. So ourquestion is:

When is θ homological?

Remark. If θ: homolog., then ∃ a recollement

D(StR T) // D(B) //ff

xx

D(R)ee

yy




Exact pairs



Condition for homological univ. localizations

Theorem

Given an exact pair (λ,µ) with λ homolog. ⇒TFAE:

(1) θ is homological ring epi.

(2) TorRj (T,S) = 0 for all j > 0.




Exact pairs



Homological subcategories

Definition

A full triang. subcat. T of D(B) is calledhomological if ∃ a homological ring epi λ : B→ Csuch that the restriction is a triangle equivalencefrom D(C) to T .

When is Ker(T⊗LB−) homological in D(B)?




Exact pairs



Condition for homological subcategories

Theorem

T : good n-tilting A-module, B := EndA(T).

TFAE:

(1) Ker(T⊗LB−): homological,

(2) Hi(HomA(P•,A)⊗A T) = 0 for i≥ 2.

P•: proj. resol. of T .




Exact pairs



consequences

Corollary

A: comm., AT : good n-tilt. mod. s.t.HomA(Ti+1,Ti) = 0 for 1≤ i≤ n−1 =⇒Ker(T⊗L

B−) is homolog. if and only ifproj.dim(AT)≤ 1.




Exact pairs



Counterexample

For n≥ 2, there is an n-tilting A-module T suchthat Ker(T⊗L

B−) is not homological. Thus thereis no homol. ring epi B→ C such that thefollowing recollement exists:

D(C) // D(B)j! //

cc

{{

D(A)cc

{{

j! := T⊗LB−




Exact pairs



open question

Open questions:

(1) What should be the replacement of HappelTheorem for inf. g. tilt. modules?(2) Find more conditions for T , such thatKer(T⊗L

B−) is homological.

changchang xi - universität bielefeld · changchang xi (¨ ¨ ¨ ~ ~ ~ ) ... let us look at the...

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